Surface Homeomorphisms Research Articles

On non-contractible periodic orbits and bounded deviations

We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic orbits. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic orbits, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or it has a single strong irrational dynamical direction.

Taking the high-edge route of rank-3 outer space

Principal fully irreducible outer automorphisms were introduced in [Y. Algom-Kfir, I. Kapovich and C. Pfaff, Stable strata of geodesics in outer space, Int. Math. Res. Not. 2019(14) (2019) 4549–4578] to emulate principal pseudo-Anosov surface homeomorphisms, i.e. those whose attracting and repelling invariant foliations have only 3-pronged singularities. It is proved in [Y. Algom-Kfir, I. Kapovich and C. Pfaff, Stable strata of geodesics in outer space, Int. Math. Res. Not. 2019(14) (2019) 4549–4578] that these outer automorphisms semi-mimic principal pseudo-Anosov surface homeomorphisms in important ways. We prove here which rank-3 graphs carry train track representatives of principal fully irreducible outer automorphisms. As a corollary, one obtains which simplices of rank-3 Culler–Vogtmann outer space principal axes pass through.

Surface Maps via Adaptive Triangulations

AbstractWe present a new method to compute continuous and bijective maps (surface homeomorphisms) between two or more genus‐0 triangle meshes. In contrast to previous approaches, we decouple the resolution at which a map is represented from the resolution of the input meshes. We discretize maps via common triangulations that approximate the input meshes while remaining in bijective correspondence to them. Both the geometry and the connectivity of these triangulations are optimized with respect to a single objective function that simultaneously controls mapping distortion, triangulation quality, and approximation error. A discrete‐continuous optimization algorithm performs both energy‐based remeshing as well as global second‐order optimization of vertex positions, parametrized via the sphere. With this, we combine the disciplines of compatible remeshing and surface map optimization in a unified formulation and make a contribution in both fields. While existing compatible remeshing algorithms often operate on a fixed pre‐computed surface map, we can now globally update this correspondence during remeshing. On the other hand, bijective surface‐to‐surface map optimization previously required computing costly overlay meshes that are inherently tied to the input mesh resolution. We achieve significant complexity reduction by instead assessing distortion between the approximating triangulations. This new map representation is inherently more robust than previous overlay‐based approaches, is less intricate to implement, and naturally supports mapping between more than two surfaces. Moreover, it enables adaptive multi‐resolution schemes that, e.g., first align corresponding surface regions at coarse resolutions before refining the map where needed. We demonstrate significant speedups and increased flexibility over state‐of‐the art mapping algorithms at similar map quality, and also provide a reference implementation of the method.

Topological horseshoes for surface homeomorphisms

In this work, we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy, and we obtain several applications. For homeomorphisms of the open annulus A with zero topological entropy, we show that rotation numbers exist for all points with nonempty ω-limit sets, and that if A is a generalized region of instability, then it admits a single rotation number. We also offer a new proof of a recent result of Passeggi, Potrie, and Sambarino, showing that zero entropy dissipative homeomorphisms of the annulus having a circloid as an attractor have a single rotation number. Our work also studies homeomorphisms of the sphere without horseshoes. For these maps, we present a structure theorem in terms of fixed point free invariant subannuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most two distinct periodic orbits and that, in many cases, the restriction of the homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti et al.

Topology Rule-Based Methodology for Flow Separation Analysis in Turbomachinery

Boundary-layer flow separation is a common flow feature in many engineering applications. The consequences of flow separation in turbomachinery can be disastrous in terms of performance, stability and noise. In this context, flow separation is particularly difficult to understand because of its three-dimensional and confined aspects. Analyzing the skin friction lines is one key point to understanding and controlling this phenomenon. In the case of separation, the flow at the wall agglutinates around a manifold while the fluid from the boundary layer is ejected toward the flow away from the wall. The analysis of a three-dimensional separation zone based on topology is well addressed for a simple geometry. This paper aims at providing simple rules and methods, with a clear vocabulary based on mathematical background, to conduct a similar analysis with complex turbomachinery geometry (to understand a surface with a high genus). Such an analysis relies on physical principles that help in understanding the mechanisms of flow separation on complex geometries. This paper includes numerous typical turbomachinery surfaces: the stator row, vaneless diffuser, vaned diffuser, axial rotor and shrouded and unshrouded centrifugal impeller. Thanks to surface homeomorphisms, the generic examples presented can easily be converted into realistic shapes. Furthermore, classical turbomachinery problems are also addressed, such as periodicity or rotor clearance. In the last section, the proposed methodology is conducted on a radial diffuser of an industrial compressor. The flow at the wall is extracted from LES computations. This study presents the different closed separation zones in a high-efficiency operating condition.

Curves on Non-Orientable Surfaces and Crosscap Transpositions

Let Ng,n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in Ng,n making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in Ng,n in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston’s theory of surface homeomorphisms, which were solved only on orientable surfaces before.

Conservative surface homeomorphisms with finitely many periodic points

Conservative surface homeomorphisms with finitely many periodic points

Handel’s Fixed Point Theorem: A Morse Theoretical Point of View

Michael Handel has proved in [10] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article fits into a series of articles by the author [13] and by Juliana Xavier [21, 22], where proofs were given, related to the classical Brouwer Theory, instead of the Homotopical Brouwer Theory used in the original article. Like in [13, 21] and [22], we will use “free brick decompositions” but will present a more conceptual Morse theoretical argument. It is based on a new preliminary lemma, that gives a nice “condition at infinity” for our problem.

Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

AbstractWe develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

AbstractWe show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

Inter-surface maps via constant-curvature metrics

We propose a novel approach to represent maps between two discrete surfaces of the same genus and to minimize intrinsic mapping distortion. Our maps are well-defined at every surface point and are guaranteed to be continuous bijections (surface homeomorphisms). As a key feature of our approach, only the images of vertices need to be represented explicitly, since the images of all other points (on edges or in faces) are properly defined implicitly. This definition is via unique geodesics in metrics of constant Gaussian curvature. Our method is built upon the fact that such metrics exist on surfaces of arbitrary topology, without the need for any cuts or cones (as asserted by the uniformization theorem). Depending on the surfaces' genus, these metrics exhibit one of the three classical geometries: Euclidean, spherical or hyperbolic. Our formulation handles constructions in all three geometries in a unified way. In addition, by considering not only the vertex images but also the discrete metric as degrees of freedom, our formulation enables us to simultaneously optimize the images of these vertices and images of all other points.

A condition that implies full homotopical complexity of orbits for surface homeomorphisms

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ isotopic to the identity. A set of hypotheses is presented, called a fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}$), denoted by $\widetilde{f},$ has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, for $C^{1+\unicode[STIX]{x1D716}}$ diffeomorphisms, we show the existence of rotational horseshoes having non-trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets and by periodic orbits in the rational case. Also, $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where $f$ is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

An algorithm to compute the Teichmüller polynomial from matrices

In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmuller polynomial corresponding to those surface homeomorphisms by first constructing an invariant track whose first homology group can be naturally identified with the first homology group of the surface, and computing its Alexander polynomial.

Book Review: Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory; Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions

Book Review: Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory; Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions

Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms

The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal identity isotopy such that we can give a fine descriptions of the dynamics of its transverse foliation. We will define a kind of identity isotopies: torsion-low isotopies. In particular, when $f$ is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy $I$ of $f$ is torsion-low if and only if for every point $z$ fixed along the isotopy, the (real) rotation number $\rho(I,z)$, which is well defined when one blows-up $f$ at $z$, is contained in $(-1,1)$. We will prove the existence of torsion-low maximal identity isotopies, and we will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity.

Forcing theory for transverse trajectories of surface homeomorphisms

This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel’s classification of zero entropy maps of $$\mathbb {S}^2$$ for non-wandering homeomorphisms; we show that if f is a Hamiltonian homeomorphism of the annulus, then the rotation set of f is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks–Misiurewicz conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms.

Fully essential dynamics for area-preserving surface homeomorphisms

We study the interplay between the dynamics of area-preserving surface homeomorphisms homotopic to the identity and the topology of the surface. We define fully essential dynamics and generalize the results previously obtained on strictly toral dynamics to surfaces of higher genus. Non-fully essential dynamics are, in a way, reducible to surfaces of lower genus, while in the fully essential case the dynamics is decomposed into a disjoint union of periodic bounded disks and a complementary invariant externally transitive continuum $C$. When the Misiurewicz–Ziemian rotation set has non-empty interior the dynamics is fully essential, and the set $C$ is (externally) sensitive on initial conditions and realizes all of the rotational dynamics. As a fundamental tool we introduce the notion of homotopically bounded sets and we prove a general boundedness result for invariant open sets when the fixed point set is inessential.

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [24] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an [Formula: see text] analog of the Masur–Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [16] give a strictly finer invariant in the analogous [Formula: see text] setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in [Formula: see text]. The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of [Formula: see text] are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.

Anomalous cw-expansive surface homeomorphisms

We prove that the genus two surface admits a cw-expansive homeomorphism with a fixed point whose local stable set is not locally connected.

Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms

Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_{0}$ with Lefschetz index one. Le Roux conjectured that $z_{0}$ is accumulated by periodic orbits. In this paper, we will approach Le Roux’s conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_{0}$ and if the area of $M$ is finite, then $z_{0}$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_{0}$ is the limit in the weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.